1. Data Structures
Hashing 1

2. Hashing Other search techniques require multiple comparisons
The goal of hashing is to reduce the number of comparisons to one
That is, we wish to locate the key immediately
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3. Hashing
A hash search is a search in which the key, through an algorithmic function, determines the location of the data.
Hashing is a key-to-address transformation in which the keys map to addresses in a list
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4. F(key) = address Key Address Hash Function 4
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5. 001 Harry Lee 002 Sarah Trapp 005 Vu Nguyen 007 Ray Black
Address
Mapping a Key value
Into a record location
[001] [002] [003] [004] [005] [006] [007] [099] [100]
001 Harry Lee Sarah Trapp Vu Nguyen Ray Black
100 John Adams
Hash
5 2
100
Key
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6. Synonyms
It is possible that two (or more) different key values produce the same location.
We call the set of keys that hashes to the same location in our list synonyms
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7. Collisions
If the actual data that we insert into our list contains two or more synonyms, then we will have a collision.
A collision is the event that occurs when a hashing algorithm produces an address for insertion that is already occupied.
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8. Collisions
The address produced by the hashing algorithm is called the home address.
The memory that contains all the home addresses is called the prime area.
When two keys collide at a home address, we must resolve the collision by placing one of the keys & its data at another location
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9. Searching
If we use one hashing algorithm to insert a key, we must use the same algorithm to find it!
Each calculation of an address and test for success (is the key & data in the list?) is called a probe.
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10. Looking for the ideal Hashing Function
Simple to use
Instantaneous computation
No waste of space
Symmetrical (data placement & retrieval use same mechanism)
Some examples follow
CIS265/506: Chapter 11 Hashing

11. Direct Hashing
The key is the address without any algorithmic manipulation
The supporting data structure must contain an element for every possible key.
Uses are somewhat limited.
Never a risk for synonyms (Why?)
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12. Direct Hashing Situation #1:
Let’s say we need to look at daily sales figures for one month.
We could set up an array with 31 distinct elements.
We could use the day as the key (& address) and the sales figures as the data
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13. Key
Data
… … …
Since we are dealing with “direct hashing” - the address & the key are the same
dailySales[current_day] = dailySales[current_day] + sale_amount
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14. Direct Hashing That sounds easy! What are some drawbacks?
Wasteful in terms of space (eg. SSN or CSU school ID produces a large….large…. prime area)
How do you find every possible case?
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15. Subtraction Method
If our keys were sequential, but did not start from one (or zero), we could use the subtraction method.
Example:
Keys were from 1000 to 1100.
We would subtract 1000 from the key to determine its address
Same problems & issues as the direct method
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16. Modulo-Division Method
Also known as the division-remainder method
Divides the key by the array size and uses the remainder plus one to produce the address
address = key MODULUS (listSize + 1)
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17. Modulo-Division Method
Can work with any list size
If the list size is a prime number, there will be fewer collisions (???)
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18. Modulo-Division Method - Example
If we had a need to store 300 pieces of information, we would choose an array size of 307 (the next largest prime number).
Now, assume we have key
Q. In what array location (address) is this key stored?
 307 = 395 with a remainder of 2
address = key MODULUS (listSize + 1) ?? = ( modulus 307) =
A. Address value is: 3
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19. Digit Extraction Method
Selected digits are extracted from the key and used as the address:
key  address
 394
 112
 388
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20. Mid-square Method
The key is squared and the address selected from the middle of the squared number
The full squared number may be too large for the computer !
If the key has 6 digits, the product is 12 digits, which is larger than the size of an integer in many computers
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21. Midsquare Method Example: Assume we have a 4 digit address (0000-9999)
Q. What is the address of the key 9452?
A * 9452 =
Address = 3403
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22. Folding Methods Fold Shift Method
The key value is divided in two parts
The left & right parts are shifted and added to the middle part
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23. Fold Shift Method Address Size Key 3 123456789 123 456 + 789 1368
← left
← middle
← right
the address for the key is 368
discarded
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24. Folding Methods Fold Boundary Method
The key value is divided in two parts
The left & right parts are folded and added to the middle part
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25. note that the beginning & ending number have been reversed!
Fold Boundary Method
Address Size 3
Key
note that the beginning & ending number have been reversed!
the address for the key is 764
discarded
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26. Note that the two folding hashing methods produce
different addresses !
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27. Collision Resolution
Whenever we are not using a direct one-to-one mapping of keys to addresses, there is a potential for a collision.
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28. 001 Harry Lee 002 Sarah Trapp 005 Vu Nguyen 007 Ray Black
202002
[001] [002] [003] [004] [005] [006] [007] [099] [100]
001 Harry Lee Sarah Trapp Vu Nguyen Ray Black
100 John Adams
Hash 5
Uh-oh!!
These two keys (green, purple)
hash to the same address!
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29. A little bit of theory…..
Because of the anticipatory nature of hashing algorithms, it is necessary to have some empty elements in the list
Practice says that a hashed list should never be more than 75% full
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30. LOAD FACTOR
the number of elements in the list divided by the number of physical elements allocated for the list expressed as a percentage
Assigned the symbol alpha (  )
k : the number of filled element
n : number of total elements
 = k/n * 100
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31. Clustering
As data are added to a list and collisions are resolved, some hashing algorithms tend to cause data to group within a list
The tendency of data to build up unevenly across a hashed list is called clustering.
High number of clusters causes decreased search efficiency
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32. Primary Clustering
Primary clustering occurs when data becomes clusters around a home address.
Easy to identify
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33. Secondary Clustering
Secondary clustering occurs when data becomes grouped along a collision path throughout a list
Not easy to identify
Rapidly decreases search efficiency
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34. Secondary Clustering
The data are widely distributed across the list so the list appears to be evenly distributed
If the data all lie along a well-traveled collision path, the time to locate a requested element of data can become large
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35. Secondary Clustering Example: Assume you have a group of n people.
We will use their birthdates as a key.
What size group is needed before you find two (or more) people with the same birthday (not necessarily the same year, but same date)?
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36. Secondary Clustering Factoid:
If there are more than 23 people in a group, there is a better than 50% chance that two people have the same birthday.
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37. Secondary Clustering
If we extrapolate this statistical curiosity into our hashing methods – we could say:
if we have a list of 365 empty addresses (one for each day in a non-leap year), we can expect to get a collision within the first 23 inserts 50% of the time.
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38. Open Addressing Collision Resolution Methods
Four variations on the theme:
Linear Probe
Quadratic Probe
Double Hashing
Key Offset
Collisions are resolved by placing the offending key in the prime area
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39. Linear Probe Simplest Method
If there is a collision, we add one to the address and try & insert the data in that location.
If that fails, add one & try again.
Keep going until there is success
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40. 001 Harry Lee 002 Sarah Trapp 005 Vu Nguyen 007 Ray Black
202002
[001] [002] [003] [004] [005] [006] [007] [099] [100]
001 Harry Lee Sarah Trapp Vu Nguyen Ray Black
100 John Adams
Hash 5
Collision!
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41. Linear Probe Advantages: Disadvantages Easy to implement
Data tends to remain near the home address
Disadvantages
Tends to produce primary clustering
Makes search algorithms more complex - especially after data have been deleted
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42. Quadratic Probe Similar to the linear probe
The increment is the collision probe number squared
Try (probe) number 1: add 12
Try (probe) number 2: add 22 to try #1
Try (probe) number 3: add 32 to try #2
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43. Quadratic Probe Advantage Disadvantages Eliminate primary clustering
Secondary clustering remains
Inefficient because of time required to square number
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44. Quadratic Probe Disadvantages (cont.)
Not possible to generate a new address for every element in the list
If the list size is a prime number, it is possible to reach at least half the elements in the list
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45. Double Hashing
Rather than using an arithmetic probe function (i.e. linear probe), we re-hash the address
Prevents primary clustering
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46. “Pseudo-random collision resolution”
We use a pseudo-random number generator such as:
y = ( (ax + c) modulo (listSize) ) + 1
where a, x, and c are some pre-defined numbers
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47. “Pseudorandom collision resolution”
A relatively simple solution
Once a collision occurs, there is only one collision resolution path that is followed by all the keys.
Can create significant secondary clustering
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48. Key Offset
Key offset is a double hashing method that produces different collision paths for different keys
Calculates the new address as a function of the old address and the key
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49. Key Offset Offset = [ key / listSize ]
address = ((offset + oldAddress) modulo (listSize) )+ 1
Example:
oldAddress = ( modulo 307) + 1 = 2
offset = [166702/307] = 543
address = (( ) modulo 307) + 1
= 239
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50. Linked List Resolution
A major disadvantage to open addressing is that each collision resolution increases the probability of future collisions
This is eliminated using linked lists
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51. 30451 Harry Lee 00432 Sarah Trapp 02305 Vu Nguyen 23007 Ray Black
[001] [002] [003] [004] [005] [006] [007] [306] [307]
Harry Lee Sarah Trapp Vu Nguyen Ray Black
John Adams
Peter Smith
Harry Eagle
. . .
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52. Bucket Hashing
Another way to avoid collisions is to hash to “buckets” or nodes that are large enough to hold multiple keys
Collisions are postponed until the buckets are filled
Wasteful in terms of space
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53. . . . main buckets overflow buckets 340 460 record pointer
981 record pointer record pointer 182 record pointer
record pointer
. . .
record pointer
652 record pointer record pointer record pointer
record pointer
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54. File Organizations Heap or unordered Sorted or sequential Hashed
places the records on disk in no particular order
Sorted or sequential
records order by a particular field
Hashed
Uses a hash function to determine record placement
B-Trees
Uses a tree structure to determine location
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55. Files of Unordered Records (Heap Files)
Records are placed in the file in the chronological order in which they are inserted
Inserting is very efficient
The last disk block is copied to a buffer; the new record is added; the block is rewritten to the disk
Searching requires a linear search - one record at a time
very expensive
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56. Files of Unordered Records (Heap Files)
Deletion
Find the correct block
Copy the block in to a buffer; delete the record; rewrite the block back to the disk
Leaves wasted, empty, space
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57. Files of Ordered Records (Sorted Files)
We can order records based on the value of a field in the record
Called the “ordering field”
If the ordering field is also the key field (guaranteed to be unique) we call the field the ordering key for the file
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58. Files of Ordered Records (Sorted Files)
Reading (in key order)) is very efficient
No sorting required
Finding the next record usually requires no additional block accesses
The next record is usually in the same block as the previous
Faster access when binary search techniques are used
Faster than linear search
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59. Hashed Files Called a “hashed” or “direct” file
The search condition is such that a record is found or not on a single attempt
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60. Internal Hashing
Hashing is typically implemented through an array of records
We have seen this technique already
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61. External Hashing Hashing for disk files is called external hashing
The target address space is made of “buckets”
Maps a key to a relative bucket number
A table in the file header converts the bucket number in to the corresponding disk block address
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62. External Hashing
Fastest possible access for retrieving an arbitrary record
Most hash functions do not maintain records in hash address order
Requires a pre-allocated amount of space
What happens when the file grows?
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63. Dynamic Hashing
The number of buckets is not fixed, but rather grows & shrinks as needed
Once the bucket overflows, it is split and a new bucket is created
The records are redistributed
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64. Dynamic Hashing Internal Nodes: Leaf Nodes
guide the search - left pointer = 1, right pointer = 0
Leaf Nodes
hold a pointer to a bucket - a bucket address
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65. DATA FILE BUCKETS
buckets for records whose hash values start with 000 1
buckets for records whose hash values start with 001
buckets for records whose hash values start with 01 1
buckets for records whose hash values start with 10 1
buckets for records whose hash values start with 110 1
buckets for records whose hash values start with 111 1 44